A194361 -id:A194361

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A143935

Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

+10
10

2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 3, 4, 1, 5, 3, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 3, 5, 2, 4, 5, 4, 5, 5, 3, 5, 5, 2, 6, 5, 4, 4, 4, 5, 5, 7, 5, 5, 3, 5, 6, 3, 8, 3, 4, 5, 6, 7, 5, 6, 8, 5, 4, 6, 6, 3, 7, 5, 4, 8, 5, 8, 6, 3, 7, 7, 6, 8, 7, 4, 5, 6, 5, 9, 9, 7, 6, 6, 6, 6, 7, 6, 4, 8, 5, 8, 8, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This value of K is conjectured to be the least possible such that there is at least one prime in the range n^k and (n+1)^k for all n>0 and k>=K. This value of K was found using exact interval arithmetic. For each n <= 300 and for each prime p in the range n to n^2, we computed an interval k(n,p) such that p is between n^k(n,p) and (n+1)^k(n,p). The intersection of all these intervals produces a list of 29 intervals. The last interval appears to be semi-infinite beginning with K, which is log(127)/log(16). See A143898 for the smallest number in the first interval.

My UBASIC program indicates no prime between 113.457 ... and 126.999 .... Next prime > 113 is 127. I would like someone to check this. - Enoch Haga, Sep 24 2008

It suffices to check members of floor(A002386^(1/k)). - Charles R Greathouse IV, Feb 03 2011

The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

MATHEMATICA

k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k], Floor[(n+1)^k]], PrimeQ]], {n, 150}]

With[{k=Log[16, 127]}, Table[Count[Range[Ceiling[n^k], Floor[(n+1)^k]], _?PrimeQ], {n, 110}]] (* Harvey P. Dale, Apr 03 2019 *)

CROSSREFS

Cf. A014085 (number of primes between n^2 and (n+1)^2).

Cf. A144256, A194361.

KEYWORD

nonn

AUTHOR

T. D. Noe, Sep 05 2008

EXTENSIONS

Corrected a(15) from 1 to 0 Enoch Haga, Sep 24 2008

My intention was to include the endpoints of the range. Using k=log(127)/log(16), the endpoint for n=15 is exactly 127, which is prime. - T. D. Noe, Sep 25 2008

STATUS

approved

A144256

a(n) = smallest prime in the range [n^K, (n+1)^K], where K = log(127)/log(16) = 1.74717117169304146332...

+10
4

2, 5, 7, 13, 17, 23, 31, 41, 47, 59, 67, 79, 89, 101, 127, 127, 149, 157, 173, 191, 211, 223, 241, 263, 277, 307, 317, 347, 359, 383, 409, 431, 457, 479, 499, 541, 557, 577, 607, 631, 659, 691, 719, 751, 787, 809, 839, 877, 907, 937, 967, 997, 1031, 1069

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

T. D. Noe submitted to primepuzzles.net the following conjecture #60, which is stronger than the Legendre's conjecture: For n>0 and k>=K, there is always a prime between n^k and (n+1)^k, where K = log(127)/log(16) = 1.74717117169304146332...

One could see that calculated terms for n=15 and n=16 yield the same value: 127, which make this conjecture (as originally defined) to be questionable. If this conjecture is modified to k>K, then there will be a distinct prime between 15^k and 16^k. It appears that the relatively large prime gap between 113 and 127 is the largest gap to overcome. Another way to correct/clarify the conjecture is to mention that both boundaries of the interval are included and that the same prime value may appear in two neighboring intervals. Of course the last version of the modified definition makes this conjecture to be different from the original Legendere conjecture (rather than to be an improvement of the original Legendere conjecture). [Alexander R. Povolotsky, Sep 26 2008]

The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

MATHEMATICA

k = Log[127]/Log[16]; Table[Select[Range[Ceiling[n^k], Floor[(n + 1)^k]], PrimeQ, 1][[1]], {n, 100}] (* T. D. Noe, Jan 30 2012 *)

PROG

(PARI) i=[]; for(n=1, 500, for(j=ceil(n^(log(127)/log(16))), ceil((n+1)^(log(127)/log(16))), if(isprime(j), i=concat(i, j)); if(isprime(j), break))); i

CROSSREFS

Cf. A000040, A144831, A143935.